In topology, a branch of mathematics, an extension topology is a topology placed on the disjoint union of a topological space and another set. There are various types of extension topology, described in the sections below.
Extension topology
Let X be a topological space and P a set disjoint from X. Consider in X ∪ P the topology whose open sets are of the form A ∪ Q, where A is an open set of X and Q is a subset of P.
The closed sets of X ∪ P are of the form B ∪ Q, where B is a closed set of X and Q is a subset of P.
For these reasons this topology is called the extension topology of X plus P, with which one extends to X ∪ P the open and the closed sets of X. As subsets of X ∪ P the subspace topology of X is the original topology of X, while the subspace topology of P is the discrete topology. As a topological space, X ∪ P is homeomorphic to the topological sum of X and P, and X is a clopen subset of X ∪ P.
If Y is a topological space and R is a subset of Y, one might ask whether the extension topology of Y - R plus R is the same as the original topology of Y, and the answer is in general no.
Note the similarity of this extension topology construction and the Alexandroff one-point compactification, in which case, having a topological space X which one wishes to compactify by adding a point ∞ in infinity, one considers the closed sets of X ∪ {∞} to be the sets of the form K, where K is a closed compact set of X, or B ∪ {∞}, where B is a closed set of X.
Open extension topology
Let X be a topological space and P a set disjoint from X. Consider in X ∪ P the topology whose open sets are of the form X ∪ Q, where Q is a subset of P, or A, where A is an open set of X.
For this reason this topology is called the open extension topology of X plus P, with which one extends to X ∪ P the open sets of X. As subsets of X ∪ P the subspace topology of X is the original topology of X, while the subspace topology of P is the discrete topology.
The closed sets in X ∪ P are of the form: Q, where Q is a subset of P, or B ∪ P, where B is a closed set of X. Note that P is closed in X ∪ P and X is open and dense in X ∪ P.
If Y a topological space and R is a subset of Y, one might ask whether the open extension topology of Y - R plus R is the same as the original topology of Y, and the answer is in general no.
Note that the open extension topology of X ∪ P is smaller than the extension topology of X ∪ P.
Assuming X and P are not empty to avoid trivialities, here are a few general properties of the open extension topology:[1]
If P is finite, X ∪ P is compact. So X ∪ P is a compactification of X in that case.
X ∪ P is connected.
If P has a single point, X ∪ P is ultraconnected.
For a set Z and a point p in Z, one obtains the excluded point topology construction by considering in Z the discrete topology and applying the open extension topology construction to Z - {p} plus p.
Closed extension topology
Let X be a topological space and P a set disjoint from X. Consider in X ∪ P the topology whose closed sets are of the form X ∪ Q, where Q is a subset of P, or B, where B is a closed set of X.
For this reason this topology is called the closed extension topology of X plus P, with which one extends to X ∪ P the closed sets of X. As subsets of X ∪ P the subspace topology of X is the original topology of X, while the subspace topology of P is the discrete topology.
The open sets of X ∪ P are of the form Q, where Q is a subset of P, or A ∪ P, where A is an open set of X. Note that P is open in X ∪ P and X is closed in X ∪ P.
If Y is a topological space and R is a subset of Y, one might ask whether the closed extension topology of Y - R plus R is the same as the original topology of Y, and the answer is in general no.
Note that the closed extension topology of X ∪ P is smaller than the extension topology of X ∪ P.
For a set Z and a point p in Z, one obtains the particular point topology construction by considering in Z the discrete topology and applying the closed extension topology construction to Z - {p} plus p.
Notes
Steen & Seebach, p. 48
References
Steen, Lynn Arthur; Seebach, J. Arthur Jr. (1995) [1978], Counterexamples in Topology (Dover reprint of 1978 ed.), Berlin, New York: Springer-Verlag, ISBN 978-0-486-68735-3, MR 0507446
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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